Binomial distribution: P(X = k) and P(X ≤ k)
The binomial distribution counts successes in n independent trials with the same success probability p.
Formulas
- P(X = k) = C(n, k) · pᵏ · (1 − p)ⁿ⁻ᵏ
- P(X ≤ k) = P(X = 0) + … + P(X = k)
- E(X) = n · p, σ = √(n · p · (1 − p))
Your turn
Three problems on this topic. Comma or point, fractions like 1/6 work too.
Task 1 of 3Binomial distribution
Task
8 parts are checked; each is faulty with probability 0.50. What is P(at most 2 faulty)?
P(X ≤ 2) (4 decimal places)
Comma or point; fractions like 1/6 work too.
Worked example
A multiple-choice question is guessed correctly with probability 0.20. 10 questions are guessed. What is P(exactly 4 correct)?
Worked solution
- 1.X ~ B(n = 10; p = 0.20)
- 2.P(X = 4) = (10 über 4) · 0.20^4 · 0.80^6
- 3.= 210 · 0.001600 · 0.262144 = 0.0881
Typical mistake: Forgetting the binomial coefficient.
10 parts are checked; each is faulty with probability 0.50. What is P(at most 3 faulty)?
Worked solution
- 1.X ~ B(10; 0.50)
- 2.P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
- 3.= 0.0010 + 0.0098 + 0.0439 + 0.1172 = 0.1719
Typical mistake: Leaving out P(X = 0) for "at most k".
Frequently asked questions
How do I work out "at least k"?
Use the complement: P(X ≥ k) = 1 − P(X ≤ k − 1).